Question

show
\[[(p\rightarrow q)\land (q\rightarrow r)]\rightarrow (p\rightarrow r)\] is a tautology, without using truth tables

Answer 1

Seems like transitivity.

Answer 2

-[(p>q)n(q>r)] v (p>r)

-(p>q) v -(q>r) v (p>r)

-(-pvq) v -(-qvr) v (-pvr)

(pn-q) v (qn-r) v -p v r

Answer 3

it is entirely obvious, but somehow i get messed up at the last step

Answer 4

ok so i am fine at
\[(p\land \lnot q)\lor (q\land \lnot r)\lor (\lnot p \land r)\]

Answer 5

(pn-q) v (qn-r)
v-p vr
--------------
(pv-p) n (-qv-p) v (qvr) n (-rvr)

T n (-qv-p) v (qvr) n T

(Tn-q) v (Tn-p) v (qnT) v (rnT)

-q v-p v q v r

(-qvq) v-p v r

T v ..... = T

Answer 6

Can't you just write that it represents a syllogism?

Answer 7

@amistre, you lost me on the last line

Answer 8

yeah i am trying to do it amistre way

Answer 9

since we got all ors, and a T; the rest dont matter

Answer 10

T or ? or ? or ? = T

Answer 11

that is fine , but what is this
(pn-q) v (qn-r)
v-p vr
?

Answer 12

oring -p to the first bit and r to the last bit

i stacked them vertical; helps me see whats going on

Answer 13

ok that is the line i need to get starting here
\[(p\land \lnot q)\lor (q\land \lnot r)\lor (\lnot p \land r)\]

Answer 14

\[(p\land \lnot q)\lor (q\land \lnot r)\lor (\lnot p \lor r)\]

Answer 15

a -> changes to "v" not "n"

Answer 16

p -> r :: -p v r

Answer 17

ah ok so we have
\[(p\land \lnot q)\lor (q\land \lnot r)\lor (\lnot p \lor r)\] and then
\[(p\land \lnot q)\lor (q\land \lnot r)\lor \lnot p \lor r\] and then we are going to commute to
\[\lnot p \lor (p \land \lnot q) \lor (q \land \lnot r) \lor q\]

Answer 18

yep

Answer 19

not a vq, vr

Answer 20

good, and from there its cake walk; hot coals cake walk, but cake walk nonetheless :)

Answer 21

\[(\lnot p \lor p )\land (\lnot p \lor \lnot q)\lor (q \lor r) \land (\lnot r \lor r)\]

Answer 22

i think that is right now yes?

Answer 23

yep; then we can T off the sides and distribute thru the middles

which cancels out the ts

Answer 24

god this makes my head spin. thanks

Answer 25

yeah, i blame guass lol

Answer 26

well i tried to do this on the fly and got hopelessly stuck, then i tried it again and got stuck again not on the fly. thanks for straightening it out!